Two-dimensional Scatterers Research Articles

Time-domain multiple traces boundary integral formulation for acoustic wave scattering in 2D

We present a novel computational scheme to solve acoustic wave transmission problems over two-dimensional composite scatterers, i.e. penetrable obstacles possessing junctions or triple points. The continuous problem is cast as a local multiple traces time-domain boundary integral formulation. For discretization in time and space, we resort to convolution quadrature schemes coupled to a non-conforming spatial spectral discretization based on second kind Chebyshev polynomials displaying fast convergence. Computational experiments confirm convergence of multistep and multistage convolution quadrature for a variety of complex domains.

Low Frequency Attenuation Characteristics of Two-Dimensional Hollow Scatterer Locally Resonant Phonon Crystals.

The finite element method (FEM) was applied to study the low frequency band gap characteristics of a designed phonon crystal plate formed by embedding a hollow lead cylinder coated with silicone rubber into four epoxy resin short connecting plates. The energy band structure, transmission loss and displacement field were analyzed. Compared to the band gap characteristics of three traditional phonon crystal plates, namely, the square connecting plate adhesive structure, embedded structure and fine short connecting plate adhesive structure, the phonon crystal plate of the short connecting plate structure with a wrapping layer was more likely to generate low frequency broadband. The vibration mode of the displacement vector field was observed, and the mechanism of band gap formation was explained based on the spring mass model. By discussing the effects of the width of the connecting plate, the inner and outer radii and height of the scatterer on the first complete band gap, it indicated that the narrower the width of the connecting plate, the smaller the thickness; the smaller the inner radius of the scatterer, the larger the outer radius; and the higher the height, the more conducive it is to the expansion of the band gap.

A hybrid technique for the analysis of two-dimensional scattering of harmonic waves by a penetrable inhomogeneous object

ABSTRACT This paper presents a hybrid technique for obtaining a numerical solution to the problem of scattering of the harmonic electromagnetic waves on a two-dimensional inhomogeneous scatterer. The scatterer is an infinite rectangular magneto-dielectric cylinder with an inhomogeneous cross section. Using the integral equation approach, the boundary conditions of the problem are reformulated in the form of integral equation of Fredholm of the second kind. The appropriate fast numerical algorithm corresponding to the problem is obtained by using the finite difference method. The results of numerical simulations obtained by implementing the proposed algorithm have been compared with the results obtained by using two electromagnetic software based on the different numerical methods. Some physical analysis of the above numerical results has been made in this study with a viewpoint of possible application.

Spectral Galerkin boundary element methods for high-frequency sound-hard scattering problems

This paper is concerned with the design of two different classes of Galerkin boundary element methods for the solution of high-frequency sound-hard scattering problems in the exterior of two-dimensional smooth convex scatterers. We prove in this paper that both methods require a small increase (in the order of \(k^\epsilon \) for any \(\epsilon > 0\)) in the number of degrees of freedom to guarantee frequency independent precisions with increasing wavenumber k. In addition, the accuracy of the numerical solutions are independent of frequency provided sufficiently many terms in the asymptotic expansion are incorporated into the integral equation formulation. Numerical results validating \(\mathcal {O}(k^\epsilon )\) algorithms are presented.

Galerkin Boundary Element Methods for High-Frequency Multiple-Scattering Problems

We consider high-frequency multiple-scattering problems in the exterior of two-dimensional smooth scatterers consisting of finitely many compact, disjoint, and strictly convex obstacles. To deal with this problem, we propose Galerkin boundary element methods, namely the frequency-adapted Galerkin boundary element methods and Galerkin boundary element methods generated using frequency-dependent changes of variables. For both of these new algorithms, in connection with each multiple-scattering iterate, we show that the number of degrees of freedom needs to increase as $$\mathcal {O}(k^{\epsilon })$$ (for any $$\epsilon >0$$) with increasing wavenumber k to attain frequency-independent error tolerances. We support our theoretical developments by a variety of numerical implementations.

Mathematical Determination of the Fréchet Derivative with Respect to the Domain for a Fluid-Structure Scattering Problem: Case of Polygonal-Shaped Domains

The characterization of the Frechet derivative of the elasto-acoustic scattered field with respect to Lipschitz continuous polygonal domains is established. The considered class of domains is of practical interest since two-dimensional scatterers are always transformed into polygonal-shaped domains when employing finite element methods for solving direct and inverse scattering problems. The obtained result indicates that the Frechet derivative with respect to the scatterer of the scattered field is the solution of the same elasto-acoustic scattering problem but with additional right-hand side terms in the transmission conditions across the fluid-structure interface. This characterization has the potential to advance the state-of-the-art of the solution of inverse obstacle problems.

Scattering of waves by a half-space of periodic scatterers using broadband Green's function.

An efficient scatterer-free full-wave solution for plane wave scattering from a half-space of two-dimensional (2D) periodic scatterers is derived using broadband Green's function. The Green's function is constructed using band solutions of the infinite periodic structure, and it satisfies boundary conditions on all the scatterers. A low wavenumber extraction technique is applied to the Green's function to accelerate the convergence of the modal expansion. This facilitates the Green's function with low wavenumber extraction (BBGFL) to be evaluated over a broadband as the modal solutions are independent of wavenumber. Coupled surface integral equations (SIE) are constructed using the BBGFL and the free-space Green's function respectively for the two half-spaces with unknowns only on the interface. The method is distinct from the effective medium approach which represents the periodic scatters with an effective medium. This new approach provides accurate near-field solutions around the interface with localized field patterns useful for surface plasmon polaritons and topological edge states examinations.

Algorithm 975

The T-matrix (TMAT) of a scatterer fully describes the way the scatterer interacts with incident fields and scatters waves, and is therefore used extensively in several science and engineering applications. The T-matrix is independent of several input parameters in a wave propagation model and hence the offline computation of the T-matrix provides an efficient reduced order model (ROM) framework for performing online scattering simulations for various choices of the input parameters. The authors developed and mathematically analyzed a numerically stable formulation for computing the T-matrix (J. Comput. Appl. Math. 234 (2010), 1702--1709). The TMATROM software package provides an object-oriented implementation of the numerically stable formulation and can be used in conjunction with the user’s preferred forward solver for the two-dimensional Helmholtz model. We compare TMATROM with standard methods to compute the T-matrix for a range of two-dimensional test scatterers with large aspect ratios and acoustic sizes. Our numerical results demonstrate the robust numerical stability of the TMATROM implementation, even with scatterers for which the standard methods are numerically unstable. The efficiency and flexibility of the TMATROM software package to handle a wide range of two-dimensional scatterers with various shapes and material properties are also demonstrated.

Shape recognition of acoustic scatterers using the singularity expansion method

Acoustic target recognition for two-dimensional (2D) acoustic scatterers is investigated using the singularity expansion method (SEM). Based on the Watson transformation series of the scattering field, the SEM poles can be calculated and their physical interpretation given, along with the exact normal mode for any acoustic scattering problem. Typical oscillatory phenomena appear as a series of damped sinusoidal signals in the time domain and as a standing-wave distribution in the space. These external oscillation modes are associated with the SEM poles. We note that the positions of these poles in the complex frequency plane are uniquely determined by the shape and flexible characteristics of the target regardless of the waveforms and positions of the incident signals. We then infer that SEM poles can be used as the characteristic parameters for target shape recognition. The relationship between the positions of SEM poles and the geometrical characters of 2D scatterers has been established not only for cylinders but also for other general 2D scatterers. The new method and the related calculation results provide an effective way to perform shape recognition using an acoustic scattering field, with potential applications in non-destructive testing and acoustic imaging.

Eigenvalues of Šeba billiards with localization of low-energy eigenfunctions

We study the localization of eigenfunctions produced by a point scatterer on a thin rectangle. We find an explicit set of eigenfunctions localized to part of the rectangle by showing that the one-dimensional Schrödinger operator with a Dirac delta potential asymptotically governs the spectral properties of the two-dimensional point scatterer. We also find the rate of localization in terms of the aspect ratio of the rectangle. In addition, we present numerical results regarding the asymptotic behavior of the localization.

Integral equations requiring small numbers of Krylov-subspace iterations for two-dimensional smooth penetrable scattering problems

This paper presents a class of boundary integral equations for the solution of problems of electromagnetic and acoustic scattering by two-dimensional homogeneous penetrable scatterers with smooth boundaries. The new integral equations, which, as is established in this paper, are uniquely solvable Fredholm equations of the second kind, result from representations of fields as combinations of single and double layer potentials acting on appropriately chosen regularizing operators. As demonstrated in this text by means of a variety of numerical examples (that resulted from a high-order Nyström computational implementation of the new equations), these “regularized combined equations” can give rise to important reductions in computational costs, for a given accuracy, over those resulting from previous iterative boundary integral equation solvers for transmission problems.

Scattering by multiple cylinders buried in a lossy half space at oblique incidence

The scattering characteristics of an obstacle are a function of its size, shape, and refractive index, as well as the refractive index of the host medium. For two-dimensional scatterer such as an infinite cylinder, they are also dependent on the incidence direction. The scattering problem is two-dimensional if the incident wave propagates perpendicular to the axis of the cylinder, whereas the problem becomes three-dimensional when the incident direction is inclined from the cylinder axis. The latter case of oblique incidence at a lossy half space containing multiple infinite cylinders is treated in this paper. The theoretical treatment utilizes Hertz potentials to formulate the propagation of inhomogeneous waves in the lossy medium, reflection of depolarized scattered waves at the half space interface, and transmission of scattered waves from within the half space. Analytical formulas are derived for the electromagnetic fields and Poynting vector of the scattered radiation emerging from the half space. Numerical examples are presented to illustrate scattering by multiple dissimilar infinite cylinders buried in a lossy half space irradiated by an obliquely incident plane wave.

Bayesian Compressive Sensing Approaches for the Reconstruction of Two-Dimensional Sparse Scatterers Under TE Illuminations

In this paper, the reconstruction of sparse scatterers under multiview transverse-electric illumination is dealt with. Starting from a probabilistic formulation of the “inverse source” problem, two Bayesian compressive sensing approaches are introduced. The former is a suitable extension of the single-task method presented earlier for the transverse-magnetic scalar case, while the other exploits an innovative multitask implementation to take into account the relationships among the “contrast currents” at the different probing views. Representative numerical results are discussed to assess, also comparatively, the numerical efficiency, the accuracy, and the robustness of the proposed approaches.

Solving Inverse Scattering Problems Based on Truncated Cosine Fourier and Cubic B-Spline Expansions

In this paper, a comparison between truncated cosine Fourier expansion (TCFE) and cubic B-spline expansion (CBSE) for representation of the unknown scatterer properties in solving inverse scattering problems is presented. In this comparison, the efficiency of both aforementioned expansion techniques is examined for permittivity and conductivity profile reconstruction problems. The study is carried out by converting the reconstruction problem to an optimization one and using the finite difference time domain (FDTD) method as forward electromagnetic (EM) solver and the differential evolution (DE) technique as global optimizer. The main benefit of the expansion representations of the unknown properties is the reduction of the ill-posedness, which is achieved by decreasing the number of unknowns of the inverse problem. The comparison is done under the same conditions of the number of population and optimization iterations. Numerical results related to the reconstruction of one-dimensional (1D) and two-dimensional (2D) scatterers indicate that both expansion methods are reliable tools for inverse scattering applications. It is shown that the use of the CBSE results in faster convergence of the reconstruction process compared to the TCFE. However, the TCFE gives more accurate reconstruction especially in the edges of scatterers. Both expansion techniques are robust against the presence of noise in the measurements.

A finite-element method for solving a scalar problem of diffraction by two-dimensional scatterers of complex structure

A scalar problem of diffraction by two-dimensional domains of complex form is solved by the finite-element method. The good agreement of the numerical results with the exact solution to the problem of plane wave diffraction by an infinite cylinder is demonstrated. In conclusion, examples of the capabilities of the program are demonstrated.

A fast numerical method for analysis of one- and two-dimensional electromagnetic scattering using a set of cardinal functions

Since many problems in one- and two-dimensional scattering from perfectly conducting bodies can be modeled by linear Fredholm integral equations, the main focus of this paper is to present a fast numerical method for solving them. The method uses Whittaker cardinal functions as a set of basis functions. By using the properties of these cardinal functions together with an appropriate quadrature rule, the computational cost of the method becomes low and the calculations are done quickly. This is due to the fact that the method uses just sampling of functions instead of integration. To show computational efficiency of this approach, some practical one- and two-dimensional scatterers are analyzed by it and the results are compared with those of two other numerical methods.

Numerical expansion-iterative method for analysis of integral equation models arising in one- and two-dimensional electromagnetic scattering

Most integral equations of the first kind are ill-posed, and obtaining their numerical solution needs often to solve a linear system of algebraic equations of large condition number. So, solving this system may be difficult or impossible. Since many problems in one- and two-dimensional scattering from perfectly conducting bodies can be modeled by Fredholm integral equations of the first kind, this paper presents an effective numerical expansion-iterative method for solving them. This method is based on vector forms of block-pulse functions. By using this approach, solving the first kind integral equation reduces to solve a recurrence relation. The approximate solution is most easily produced iteratively via the recurrence relation. Therefore, computing the numerical solution does not need to directly solve any linear system of algebraic equations and to use any matrix inversion. Also, the method practically transforms solving of the first kind Fredholm integral equation which is inherently ill-posed into solving second kind Fredholm integral equation. Another advantage is low cost of setting up the equations without applying any projection method such as collocation, Galerkin, etc. To show convergence and stability of the method, some computable error bounds are obtained. Test problems are provided to illustrate its accuracy and computational efficiency, and some practical one- and two-dimensional scatterers are analyzed by it.

Target statistical correlation characteristic for spatial–frequency jointly diversity multiple-input multiple-output radar

In diversity multiple-input multiple-output (MIMO) radar system, the mutual correlation of target echo signals in its diversity channels is an important concern for designing signal processing algorithms. The target temporal, spatial and frequency correlation characteristic is studied at the background of a diversity MIMO radar with multiple widely separated radar sites all capable of working at multiple carrier frequencies. With a two-dimensional round-shaped scatterers centre target model, the correlation coefficient of target echo signals in any type of diversity channel pair (DCP) is proved to be a function of the target size and the equivalent frequency interval. The theoretical correlation coefficient is verified by numerical experiments. The correlation coefficient of target echo signals often depends on the target location, which is shown for two typical DCP types.

Numerical method for analysis of one- and two-dimensional electromagnetic scattering based on using linear Fredholm integral equation models

Since many problems in one- and two-dimensional electromagnetic scattering can be modeled by linear Fredholm integral equations of the first or the second kind, the main focus of this paper is to present an effective numerical method for solving them. This method is based on vector forms of block-pulse functions. By using this approach, solving the linear Fredholm integral equation reduces to solve a linear system of algebraic equations. The advantages of the proposed method are low cost of setting up the equations without applying any projection method such as collocation, Galerkin, etc., setting up a linear system of algebraic equations of appropriate condition number, and good accuracy. Test problems are provided to illustrate its accuracy and computational efficiency, and some practical one- and two-dimensional scatterers are analyzed by it.

An optimization method for acoustic inverse obstacle scattering problems with multiple incident waves

The inverse problem considered in this article is to determine the shape of a two-dimensional time-harmonic acoustic scatterer with Dirichlet boundary conditions from the knowledge of some far field patterns. Based on the optimization method due to Kirsch and Kress for the inverse scattering problem, we propose a new scheme by reformulating the cost functional via a technique of piecewise integration with respect to incident directions. Convergence analysis of this method is given. Numerical experiments show that our method accelerates the computations without losing the accuracy of the reconstructions for the full-aperture problems. The method is extended to the limited-aperture case by weighting the total fields with special factors. Numerical examples for limited-aperture problems are also presented which show that our method produces satisfactory results efficiently in the illuminated regions.